task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | Find the smallest positive integer $y$ which satisfies the congruence $56y + 8 \equiv 6 \pmod {26}$. | 6 | 33 | 1 |
math | Given a sequence $\left\{a_n\right\}$ with the sum of the first $n$ terms $S_n=n^2+2n$, and a sequence $\left\{b_n\right\}$ satisfying $3^{n-1}b_n=a_{2n-1}$,
$(1)$ Find $a_n, b_n;$
$(2)$ Let $T_n$ be the sum of the first $n$ terms of the sequence $\left\{b_n\right\}$, find $T_n.$ | \dfrac{15}{2}- \dfrac{4n+5}{2\cdot3^{n-1}} | 113 | 26 |
math | Altitudes $\overline{AX}$ and $\overline{BY}$ of an acute triangle $ABC$ intersect at $H$. If $\angle BAC = 53^\circ$ and $\angle ABC = 82^\circ$, then what is $\angle CHY$? | 45^\circ | 60 | 4 |
math | Compute $(\cos 195^\circ + i \sin 195^\circ)^{36}.$ | -1 | 26 | 2 |
math | A certain shopping mall held a promotion during the "May Day" period, and according to the total amount of one-time purchases by customers based on the marked prices of the goods, corresponding discount methods were stipulated:<br/>① If the total amount does not exceed $500, no discount will be given;<br/>② If the tota... | 910 | 206 | 3 |
math | Three sides \(OAB, OAC\) and \(OBC\) of a tetrahedron \(OABC\) are right-angled triangles, i.e. \(\angle AOB = \angle AOC = \angle BOC = 90^\circ\). Given that \(OA = 7\), \(OB = 2\), and \(OC = 6\), find the value of
\[
(\text{Area of }\triangle OAB)^2 + (\text{Area of }\triangle OAC)^2 + (\text{Area of }\triangle OBC... | 1052 | 136 | 4 |
math | Given the function $f(x)=A\cos(\omega x + \varphi)$ defined on $\mathbb{R}$, with $A>0$, $\omega>0$, and $|\varphi| \leq \frac{\pi}{2}$, satisfies the following conditions: the maximum value is 2, the distance between two adjacent minimal points of the graph is $\pi$, and the graph of $f(x)$ is symmetric about the poin... | \left[-1, \frac{2\sqrt{2} + 1}{2}\right] | 228 | 22 |
math | In the geometric sequence $\{a_n\}$, where $a_2a_5=-\frac{3}{4}$ and $a_2+a_3+a_4+a_5=\frac{5}{4}$, calculate $\frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+\frac{1}{a_5}$. | -\frac{5}{3} | 87 | 7 |
math | In the polar coordinate system, the curve $\rho=4\sin \left( \theta- \frac{\pi}{3} \right)$ is symmetric about what axis? | \frac{5\pi}{6} | 36 | 9 |
math | Find the largest positive integer \(n\) for which there exist \(n\) finite sets \(X_{1}, X_{2}, \ldots, X_{n}\) with the property that for every \(1 \leq a<b<c \leq n\), the equation \(\left|X_{a} \cup X_{b} \cup X_{c}\right|=\lceil\sqrt{a b c}\rceil\) holds. | 4 | 94 | 1 |
math | Let set $A = \{a, b\}$, and set $B = \{c, d, e\}$. Then, the number of mappings that can be established from $A$ to $B$ is ____, and the number of mappings that can be established from $B$ to $A$ is ____. | 9, 8 | 69 | 4 |
math | A truck travels $\frac{b}{4}$ feet every $t$ seconds. There are $2$ feet in a yard. Calculate the number of yards the truck travels in $5$ minutes. | \frac{37.5b}{t} | 41 | 11 |
math | Consider four functions, each represented by a graph, labelled (2) through (5). The domain of function (3) is now $$\{-6, -5, -4, -3, -2, -1, 0, 1, 2, 3\}.$$ Determine the product of the labels of the functions which are invertible. Assume:
- Function (2) is \( y = x^3 - 2x \) on \([-2, 4]\),
- Function (3) has discret... | 120 | 245 | 3 |
math | (1) Calculate the value of $0.0081^{ \frac {1}{4}}+(4^{- \frac {3}{4}})^2+(\sqrt {8})^{- \frac {4}{3}}-16^{0.75}$;
(2) Given $log_{32}9=p$, $log_{27}25=q$, express $lg5$ in terms of $p$ and $q$. | \frac {15pq}{15pq+4} | 96 | 13 |
math | Let event $A$ be "The line $ax - by = 0$ intersects the circle $(x - 2\sqrt{2})^2 + y^2 = 6$".
(1) If $a$ and $b$ are the numbers obtained by rolling a dice twice, find the probability of event $A$.
(2) If the real numbers $a$ and $b$ satisfy $(a - \sqrt{3})^2 + (b - 1)^2 \leq 4$, find the probability of event $A$. | \frac{1}{2} | 118 | 7 |
math | In triangle \(ABC\), angle \(C\) is \(75^\circ\), and angle \(B\) is \(60^\circ\). The vertex \(M\) of the isosceles right triangle \(BCM\) with hypotenuse \(BC\) is located inside triangle \(ABC\). Find angle \(MAC\). | 30^\circ | 68 | 4 |
math | Given that $\alpha$ and $\beta$ are acute angles, $\cos\alpha=\frac{{\sqrt{5}}}{5}$, $\cos({\alpha-\beta})=\frac{3\sqrt{10}}{10}$, find the value of $\cos \beta$. | \frac{\sqrt{2}}{10} | 61 | 11 |
math | Let event $A$ have a probability of occurring as $\frac{3}{5}$ and event $B$ have a probability of occurring as $\frac{4}{5}$. Determine the smallest interval that necessarily contains the probability $p$ that both event $A$ and event $B$ occur simultaneously.
A) $[0, \frac{1}{2}]$
B) $[\frac{1}{4}, \frac{3}{4}]$
C) $[... | [\frac{2}{5}, \frac{3}{5}] | 140 | 14 |
math | Using the digits 1, 2, 3, 4 only once to form a 4-digit number, how many of them are divisible by 11? | 8 | 35 | 1 |
math | Given vectors $\overrightarrow{a}=(2m+1,3,m-1)$ and $\overrightarrow{b}=(2,m,-m)$, and $\overrightarrow{a}\parallel \overrightarrow{b}$, determine the value of the real number $m$. | -2 | 58 | 2 |
math | What is the greatest integer less than 200 for which the greatest common factor of that integer and 72 is 9? | 189 | 28 | 3 |
math | If the function $f(x) = x^3 + ax$ has two extreme points on $\mathbb{R}$, then the range of the real number $a$ is ______. | a < 0 | 39 | 4 |
math | Calculate the domain of the function \( f(x) = \log_5(\log_3(\log_2(x^2))) \). | (-\infty, -\sqrt{2}) \cup (\sqrt{2}, \infty) | 29 | 22 |
math | Among the following propositions, the correct ones are __________.
(1) The regression line $\hat{y}=\hat{b}x+\hat{a}$ always passes through the center of the sample points $(\bar{x}, \bar{y})$, and at least through one sample point;
(2) After adding the same constant to each data point in a set of data, the variance ... | (2)(6)(7) | 292 | 7 |
math | If the radius of the top base of a truncated cone is $5$, the radius of the bottom base is $R$, and a section (a plane parallel to the top and bottom bases and equidistant from them) divides the truncated cone into two parts with a ratio of side areas of $1:2$, then $R=\_\_\_\_\_\_.$ | 25 | 75 | 2 |
math | Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. Given that $a_1=2$, and the sequence $\left| \sqrt{S_n} \right|$ is also an arithmetic sequence, find the value of $a_{26}$. | 102 | 66 | 3 |
math | Find the degree measure of the smallest positive angle $\theta$ for which
\[\sin 10^\circ = \cos 40^\circ - \cos \theta.\] | 30^\circ | 38 | 4 |
math | Given the function $f(x) = \ln x - \frac{1}{2}a(x-1)$ $(a \in \mathbb{R})$.
(Ⅰ) If $a=-2$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$.
(Ⅱ) If the inequality $f(x) < 0$ holds for any $x \in (1, +\infty)$, determine the range of the real number $a$. | [2, +\infty) | 114 | 8 |
math | A book with 53 pages numbered 1 to 53 has its pages renumbered in reverse, from 53 to 1. For how many pages do the new page number and old page number share the same units digit? | 11 | 50 | 2 |
math | Form a four-digit number using the digits 1, 2, and 3. Each digit must be used at least once, and the same digit cannot be adjacent to itself. Determine the total number of such four-digit numbers that can be formed. | 18 | 51 | 2 |
math | Solve for the ordered pair $(x,y)$ that satisfies the simultaneous equations:
\begin{align*}
3x - 4y &= -2, \\
4x + 5y &= 23.
\end{align*} | \left(\frac{82}{31}, \frac{77}{31}\right) | 50 | 22 |
math | A positive integer \( \overline{ABC} \), where \( A, B, C \) are digits, satisfies
\[
\overline{ABC} = B^{C} - A
\]
Find \( \overline{ABC} \). | 127 | 54 | 3 |
math | Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $... | 160 | 248 | 3 |
math | Find the remainder when \(x^4 + 1\) is divided by \(x^2 - 2x + 4.\) | -8x + 1 | 28 | 6 |
math | Given two circles $O_1: x^2 + y^2 = 5$ and $O_2: (x-m)^2 + y^2 = 5$ ($m \in \mathbb{R}$) intersect at points A and B, and the tangents to the circles at point A are perpendicular to each other, then the length of segment AB is \_\_\_\_\_\_. | \sqrt{10} | 86 | 6 |
math | Given the sequence: $$( \frac{1}{1}), ( \frac{1}{2}, \frac{2}{1}), ( \frac{1}{3}, \frac{2}{2}, \frac{3}{1}), ( \frac{1}{4}, \frac{2}{3}, \frac{3}{2}, \frac{4}{1}), \ldots, ( \frac{1}{n}, \frac{2}{n-1}, \frac{3}{n-2}, \ldots, \frac{n-1}{2}, \frac{n}{1})$$, denote the elements of the sequence as: $a_1, a_2, a_3, a_4, a_5... | 7 | 178 | 1 |
math | Non-zero vectors \(\vec{a}\) and \(\vec{b}\) satisfy \(|\vec{a}| = |\vec{b}| = |\vec{a} - \vec{b}|\). The angle between \(\vec{a}\) and \(\vec{a} + \vec{b}\) is equal to? | 30^\circ | 73 | 4 |
math | Given circle C: x² + y² - 2x = 0, calculate the distance from the center of circle C to the origin O. | 1 | 31 | 1 |
math | Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{b}|=2|\overrightarrow{a}|=1$, and $\overrightarrow{a}$ is perpendicular to $(\overrightarrow{a}+\overrightarrow{b})$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | \frac{2\pi}{3} | 80 | 9 |
math | A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, given that according to regulations, a stove should not be located closer than 75 cm to the main gas pipe. | \frac{1}{16} | 54 | 8 |
math | A rectangular wooden block is 6 inches long, 3 inches wide, and 2 inches high. The block is painted blue on all six sides and then cut into 1 inch cubes. How many of the cubes each have a total number of blue faces that is an even number? | 20 | 58 | 2 |
math | If the 200th day of some year is a Sunday and the 100th day of the following year is also a Sunday, what day of the week was the 300th day of the previous year? Provide the answer as the number of the day of the week (if Monday, then 1; if Tuesday, then 2, etc.). | 1 | 78 | 1 |
math | If $|a|=6$, $|b|=2$, and $a+b \gt 0$, then the value of $a-b$ is ______. | 4 \text{ or } 8 | 33 | 8 |
math | You want to sort the numbers 54321 using block moves. In other words, you can take any set of numbers that appear consecutively and put them back in at any spot as a block. What is the minimum number of block moves necessary to get 12345? | 3 | 62 | 1 |
math | For a number $x$, we use $\left(x\right]$ to represent the largest integer less than $x$, for example: $\left(1.6\right]=1$, $\left(-4\right]=-5$.
① Fill in the blanks: $(0 ]=\_\_\_\_\_\_,(-2023 ]=\_\_\_\_\_\_$;
② If $|\left(x\right]-3|=6$, then the range of values for $x$ is ______. | 9 < x \leqslant 10 \text{ or } -3 < x \leqslant -2 | 106 | 27 |
math | Given the function $f(x)=\cos 2x+a\sin x+b$ where $a \lt 0$.
$(1)$ If the maximum value of $f(x)$ is $\frac{9}{8}$ and the minimum value is $-2$ when $x\in R$, find the values of real numbers $a$ and $b$.
$(2)$ If $a=-2$ and $b=1$, and the function $g(x)=m\sin x+2m$ where $x\in[\frac{\pi}{6},\frac{2\pi}{3}]$, and $... | (-\infty, -\frac{2}{3}) | 151 | 13 |
math | A circle of radius 5 is centered at point $B$. An equilateral triangle with a vertex at $B$ has a side length of 10. Find the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle. | 25(\pi - \sqrt{3}) | 67 | 10 |
math | If $x > -1$, then when $x=$_______, $f(x)=x+\frac{1}{x+1}$ has its minimum value, which is _______. | 1 | 38 | 1 |
math | Write down an integer that is greater than $\sqrt{2}$ and less than $\sqrt{17}$ ____. | 3 | 24 | 1 |
math | What is the maximum possible number of rays in a plane emanating from a single point and forming pairwise obtuse angles? | 4 | 24 | 1 |
math | Let $n$ be an odd integer with exactly 12 positive divisors. Find the number of positive divisors of $27n^3$. | 256 | 32 | 3 |
math | Calculate the result of $\sin 5^{\circ}\cos 55^{\circ}-\cos 175^{\circ}\sin 55^{\circ}$. | \frac{\sqrt{3}}{2} | 39 | 10 |
math | In a triangle, the larger angle at the base is $45^{\circ}$, and the altitude divides the base into segments of 20 and 21. Find the length of the larger lateral side. | 29 | 45 | 2 |
math | Sides $\overline{AJ}$ and $\overline{EF}$ of regular decagon $ABCDEFGHIJ$ are extended to meet at point $Q$. What is the degree measure of angle $Q$? | 72^\circ | 43 | 4 |
math | Consider the simultaneous equations
$$
\left\{\begin{array}{l}
x y + x z = 255 \\
x z - y z = 224
\end{array}\right.
$$
Find the number of ordered triples of positive integers \((x, y, z)\) that satisfy the above system of equations. | 2 | 74 | 1 |
math | On the number line, moving point $A$ by $2$ units exactly reaches the point representing $-2$. What number does point $A$ represent? | 0 \text{ or } -4 | 33 | 8 |
math | The NIMO problem writers have invented a new chess piece called the *Oriented Knight*. This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right. How many ways can the knight move from the bottom-left square ... | 252 | 100 | 3 |
math | Let $x_1 < x_2 < x_3$ be the three real roots of the equation $\sqrt{2020} x^3 - 4041x^2 + 3 = 0$. Find $x_2(x_1+x_3)$. | 3 | 62 | 1 |
math | The magnitude of the vector $\overrightarrow{a}+2\overrightarrow{b}$ can be calculated using the given information that the angle between the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$. | 2\sqrt{3} | 75 | 6 |
math | Find all pairs of natural numbers \( a \) and \( b \) such that out of the four statements:
5) \( a^2 + 6a + 8 \) is divisible by \( b \);
6) \( a^2 + ab - 6b^2 - 15b - 9 = 0 \);
7) \( a + 2b + 2 \) is divisible by 4;
8) \( a + 6b + 2 \) is a prime number;
three are true, and one is false. | (5, 1) \text{ and } (17, 7) | 118 | 18 |
math | Given that $3\sin a-\cos a=0$ and $7\sin β+\cos β=0$, and also given that $0 < a < \frac{π}{2} < β < π$, determine the value of $2α-β$. | -\frac{3π}{4} | 56 | 8 |
math | Given two circles are internally tangent at a point, with circles centered at points $A$ and $B$ having radii $7$ and $4$ respectively, find the distance from point $B$ to the point where an internally tangent line intersects ray $AB$ at point $C$. | 4 | 60 | 1 |
math | There are \( n \) pieces of paper, each containing 3 different positive integers no greater than \( n \). Any two pieces of paper share exactly one common number. Find the sum of all the numbers written on these pieces of paper. | 84 | 49 | 2 |
math | A list of five positive integers has all of the following properties:
- The only integer in the list that occurs more than once is $7$,
- Its median is $10$,
- Its average (mean) is $12$.
What is the largest possible integer that could appear in the list? | 25 | 64 | 2 |
math | Given the sets $A=\{x| \frac {[x]-1}{x}<0\}$, $B=\{x|x^2-3x-4\leq0\}$, and $C=\{x|\log_{\frac{1}{2}}x>1\}$, and the descriptions from students A, B, and C, determine the value of $[x]$. | 1 | 85 | 1 |
math | Five people of heights 65, 66, 67, 68, and 69 inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly 1 inch taller or exactly 1 inch shorter than himself? | 14 | 69 | 2 |
math | The inclination angle of the line $\sqrt{3}x + y - 1 = 0$ can be calculated using the slope. | 120^{\circ} | 28 | 7 |
math | The set \( A \) consists of integers, with the smallest element being 1 and the largest being 100. Except for 1, every element is equal to the sum of two (not necessarily distinct) elements from the set \( A \). Find the minimum number of elements in set \( A \). | 9 | 65 | 1 |
math | How many real solutions are there for \(y\) in the following equation:
\[
(2y + 5)^2 - 7 = -|3y + 1|
\] | 2 | 40 | 1 |
math | Given two vectors $\overrightarrow{a} = \left(\frac{3}{2}, 1+\sin \alpha\right)$ and $\overrightarrow{b} = \left(1-\cos\alpha, \frac{1}{3}\right)$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the acute angle $\alpha$. | 45^{\circ} | 79 | 6 |
math | Selected Exercise $(4-4)$: Coordinate Systems and Parametric Equations
Given that in the rectangular coordinate system $xOy$, the equation of the ellipse $(C)$ is $\frac{{y}^{2}}{16}+\frac{{x}^{2}}{4}=1$. Establish a polar coordinate system with $O$ as the pole and the non-negative semiaxis as the polar axis, using th... | 9 | 188 | 1 |
math | Find the sum of all divisors \( d \) of \( N = 19^{88} - 1 \) that are of the form \( d = 2^a \cdot 3^b \) where \( a \) and \( b \) are natural numbers. | 819 | 61 | 3 |
math | Given that a normal vector of line $l$ is $\overrightarrow{n}=(\sqrt{3}, -1)$, find the size of the slope angle of line $l$. | \frac{\pi}{3} | 38 | 7 |
math | Given that the centers of two circles are both on the line $x-y+1=0$ and they intersect at two different points. If one of the intersection points is $A(-2, 2)$, then the coordinates of the other intersection point are \_\_\_\_\_\_. | (1, -1) | 60 | 6 |
math | From the following infinite list of numbers, how many are integers? $$\sqrt{1024},\sqrt[3]{1024},\sqrt[4]{1024},\sqrt[5]{1024},\sqrt[6]{1024},\ldots$$ | 3 | 66 | 1 |
math | The number of integers whose absolute value is less than 4, their sum, and their product are. | 0 | 21 | 1 |
math | Given an arithmetic sequence $\{a\_n\}$ with a common ratio of $q (q > 0)$, let $S_n$ represent the sum of its first $n$ terms. If $S_2 = 3a_2 + 2$ and $S_4 = 3a_4 + 2$, find the value of $q$. | \frac{3}{2} | 78 | 7 |
math | Jar C initially contains 6 red buttons and 12 green buttons. Michelle removes the same number of red buttons as green buttons from Jar C and places them into an empty Jar D. After the removal, Jar C is left with $\frac{3}{4}$ of its initial button count. If Michelle were to randomly choose a button from Jar C and a but... | \frac{5}{14} | 97 | 8 |
math | Triangles $PQR$ and $PRS$ are isosceles with $PQ=QR$ and $PR=RS$. Point $R$ is inside $\triangle PQR$, $\angle PQR = 50^\circ$, and $\angle PRS = 130^\circ$. What is the degree measure of $\angle QPR$? | 40^\circ | 74 | 4 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\cos C=3a\cos B-c\cos B$, $\overrightarrow{BA}\cdot \overrightarrow{BC}=2$, find the area of $\triangle ABC$. | 2\sqrt{2} | 71 | 6 |
math | Given that 10% of the students got 70 points, 25% got 80 points, 20% got 85 points, 15% got 90 points, and the rest got 95 points, calculate the difference between the mean and the median score on this exam. | 1 | 68 | 1 |
math | If $|a|=2$, $b^{2}=9$, and $a \lt b$, find the value of $a-b$. | -1 \text{ or } -5 | 29 | 9 |
math | Given a function $f(x)=\ln(\sqrt{1+x^2}-x)+\frac{2}{2^x+1}+1$, if $f(m-1)+f(1-2m) > 4$, then the range of real number $m$ is ______. | (0,+\infty) | 63 | 7 |
math | An ellipse has foci at $(3, 3)$ and $(3, 7)$, and it passes through the point $(9, -2)$. Write the equation of this ellipse in the standard form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\] where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find the ordered quadruple $(a, b, h, k)$. | \left(\frac{\sqrt{61} + \sqrt{117}}{2}, \sqrt{\left(\frac{\sqrt{61} + \sqrt{117}}{2}\right)^2 - 4}, 3, 5\right) | 113 | 59 |
math | In triangle \( ABC \), given \( a^{2} + b^{2} + c^{2} = 2\sqrt{3} \, ab \, \sin C \), find \( \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \). | \frac{3\sqrt{3}}{8} | 71 | 12 |
math | Define a sequence of integers by $T_1 = 2$ and for $n\ge2$ , $T_n = 2^{T_{n-1}}$ . Find the remainder when $T_1 + T_2 + \cdots + T_{256}$ is divided by 255.
*Ray Li.* | 20 | 79 | 2 |
math | a) In how many different ways can a convex octagon be divided into triangles by diagonals that do not intersect inside the octagon?
b) Euler's problem. In how many ways can a convex $n$-gon be divided into triangles by diagonals that do not intersect inside the $n$-gon? | \frac{(2n-4)!}{(n-1)!(n-2)!} | 65 | 20 |
math | Given the arithmetic sequence $\{a_n\}$, $a_5 > 0$ and $a_4 + a_7 < 0$, find the maximum term in the sum of the first $n$ terms $S_n$ of $\{a_n\}$. | S_5 | 58 | 3 |
math | Given \\(f(x)=x\ln x\\), \\(g(x)=x^{3}+ax^{2}-x+2\\).
\\((\\)I\\()\\) If the function \\(g(x)\\) is monotonically decreasing in the interval \\((- \dfrac {1}{3},1)\\), find the expression of function \\(g(x)\\);
\\((\\)II\\()\\) Under the condition of \\((\\)I\\()\\), find the equation of the tangent line to the gr... | [-2,+\infty) | 185 | 7 |
math | The exchange rate of the cryptocurrency Chukhoyn was one dollar on March 1, and then increased by one dollar each day. The exchange rate of the cryptocurrency Antonium was also one dollar on March 1, and then each day thereafter, it was equal to the sum of the previous day's rates of Chukhoyn and Antonium divided by th... | 92/91 | 96 | 5 |
math | Assume that $e$, $f$, $g$, and $h$ are positive integers such that $e^5 = f^4$, $g^3 = h^2$, and $g - e = 31$. Determine $h - f$. | 971 | 55 | 3 |
math | Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\begin{cases}x=2+t \\ y=kt\end{cases}$ (where $t$ is the parameter), and the parametric equation of line $l_2$ is $\begin{cases}x=-2+m \\ y= \frac{m}{k}\end{cases}$ (where $m$ is... | \sqrt{5} | 241 | 5 |
math | There is a tetrahedron with vertices $P$, $Q$, $R$, and $S$. The length of the shortest trip from $P$ to $R$ along the edges is the length of 4 edges. How many different 4-edge trips are there from $P$ to $R$? | 4 | 65 | 1 |
math | Given the sequence $\{a\_n\}$ that satisfies $a_{n+1} - a_{n} = 2$, where $n \in \mathbb{N}^*$, and $a_{3} = 3$, find $a_{1} =$\_\_\_\_\_\_ and its first $n$ terms sum $S_{n} =$\_\_\_\_\_\_. | S_{n} = n^2 - 2n | 87 | 12 |
math | Use the Euclidean algorithm to find the greatest common divisor of 5280 and 12155. | 55 | 25 | 2 |
math | For arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sums of the first $n$ terms of $\{b_n\}$ are denoted as $S_n$ and $T_n$, respectively. For all natural numbers $n$, it holds that $$\frac {S_{n}}{T_{n}} = \frac {n}{n+1}$$. Evaluate $\frac {a_{5}}{b_{5}}$. | \frac{9}{10} | 96 | 8 |
math | Given that $P$ is a moving point on the line $3x+4y+8=0$, $PA$ and $PB$ are two tangents of the circle $x^{2}+y^{2}-2x-2y+1=0$, $A$ and $B$ are the points of tangency, and $C$ is the center of the circle. Find the minimum value of the area of quadrilateral $PACB$. | 2\sqrt{2} | 97 | 6 |
math | Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$? | 6 | 67 | 1 |
math | Anna flips an unfair coin 10 times. The coin has a $\frac{1}{3}$ probability of coming up heads and a $\frac{2}{3}$ probability of coming up tails. What is the probability that she flips exactly 7 tails? | \frac{5120}{19683} | 53 | 14 |
math | Given the right triangles ABC and ABD, what is the length of segment BC, in units? [asy]
size(150);
pair A, B, C, D, X;
A=(0,0);
B=(0,15);
C=(-20,0);
D=(-45,0);
draw(A--B--D--A);
draw(B--C);
draw((0,2)--(-2,2)--(-2,0));
label("$50$", (B+D)/2, NW);
label("$25$", (C+D)/2, S);
label("$20$", (A+C)/2, S);
label("A", A, SE);... | 25 | 173 | 2 |
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